On profinite completions and canonical extensions

We show that if a variety V of monotone lattice expansions is finitely generated, then profinite completions agree with canonical extensions on V. The converse holds for varieties of finite type. This paper is dedicated to Walter Taylor. Received May 14, 2005; accepted in final form September 8, 2005.     

Natural extensions and profinite completions of algebras

This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class \({\mathcal{A} = \mathbb{ISP}(\mathcal{M})}\), where \({\mathcal{M}}\) is a set, not necessarily finite, of finite algebras, it is shown that each \({{\bf A} \in \mathcal{A}}\) embeds as a topologically dense subalgebra of a topological algebra \({n_{\mathcal{A}}({\bf A})}\) (its natural extension), and that \({n_{\mathcal{A}}({\bf A})}\) is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that \({\mathcal{M}}\) is finite and \({\mathcal{A}}\) possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply.     

MacNeille completions and canonical extensions

Let be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety is generated by an elementary class of relational structures.

Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.

     

Grothendieck''s problem on profinite completions and representations of groups

The paper is concerned with Grothendieck's problem on profinite completions of groups. The relationship of this problem to the representation theory of finitely generated groups and to the problem of arithmeticity of Platonov are treated.To Professor A. Grothendieck on the Occasion of his 60th Birthday     

Some remarks on profinite HNN extensions

We extend a construction of Higman, Neumann and Neumann [LS, IV.3.1] and show that every profinite groupG with only countably many open subgroups embeds in a 2-generated profinite groupE in which all torsion elements are conjugate to elements ofG; ifG is pro-p,E can be chosen pro-p. This answers a question of Wilson (oral communication) and generalises a result of Lubotzky and Wilson [LW].     

Profinite groups, profinite completions and a conjecture of Moore

Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ-module, whose restriction to RH is projective.Moore's conjecture (J. Pure Appl. Algebra 7(1976)287): Assume for every nontrivial element x in Γ, at least one of the following two conditions holds:

(M1)

〈x〉∩H≠{e} (in particular this holds if Γ is torsion free)

(M2)

ord(x) is finite and invertible in R.

Then M is projective as an RΓ-module.More generally, the conjecture has been formulated for crossed products R*Γ and even for strongly graded rings R(Γ). We prove the conjecture for new families of groups, in particular for groups whose profinite completion is torsion free.The conjecture can be formulated for profinite modules M over complete groups rings [[RΓ]] where R is a profinite ring and Γ a profinite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.     

A note on coring extensions

A notion of a coring extension is defined and it is shown to be equivalent to the existence of an additive functor between comodule categories that factorises through forgetful functors. This correspondence between coring extensions and factorisable functors is illustrated by functors between categories of descent data. A category in which objects are corings and morphisms are coring extensions is also introduced.

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Sunto Si fornisce una definizione per estensioni di coanelli e si dimostra l'equivalenza di tale definizione con l'esistenza di un funtore additivo tra categorie di comoduli che si fattorizzi attraverso il funtore dimenticante. Questa corrispondenza tra estensioni di coanelli e funtori fattorizzabili è illustrata da funtori tra categorie di discesa. Si introduce inoltre una categoria i cui oggetti sono coanelli e i morfismi sono estensioni di coanelli.

     

A note on end extensions

Abstract. We provide an alternative proof of a theorem of P. Clote concerning end extensions of models of -induction, for . Received: 10 November 1998     

A note on canonical functions

We construct a generic extension in which the ℵ₂nd canonical function on ℵ₁ exists. Supported by NSF and by a Fulbright grant. Publ. 378. Partially supported by the B.S.F.     

A note on quasi-Frobenius extensions

It is shown that for a quasi-Frobenius extension A of a right non-singular ring B if A is a right self-injective ring, then so is B.     

A note on strong conciseness in virtually nilpotent profinite groups

Archiv der Mathematik - A group word w is said to be strongly concise in a class $${\mathcal {C}}$$ of profinite groups if, for every group G in $${\mathcal {C}}$$ such that w takes less than...     

A note on characteristic functions and their extensions

Let f : ? → ? be the characteristic function of a probability measure. We study the following question: Is it true that for any closed interval I on ? that does not contain the origin, there exists a characteristic function g coinciding with f on I but not on the whole ??     

Matrix extensions and eigenvalue completions, the generic case

In this paper we provide new necessary and sufficient conditions for the so-called eigenvalue completion problem.

     

A note on extensions of asymptotic density

By a density we mean any extension of the asymptotic density to a finitely additive measure defined on all sets of natural numbers. We consider densities associated to ultrafilters on and investigate two additivity properties of such densities. In particular, we show that there is a density for which is complete.

     

A note on extensions of infinitary logic

We show that a strong form of the so called Lindströms Theorem [4] fails to generalize to extensions of L and L : For weakly compact there is no strongest extension of L with the (,)-compactness property and the Löwenheim-Skolem theorem down to . With an additional set-theoretic assumption, there is no strongest extension of L with the (,)-compactness property and the Löwenheim-Skolem theorem down to <.We are indebted to Lauri Hella, Tapani Hyttinen and Kerkko Luosto for useful suggestions.Research partially supported by the United States-Israel Binational Science Foundation. Publication number [ShVa:726]Research partially supported by grant 40734 of the Academy of Finland.